IB Questionbank

User interface language: English | Español

Date	May 2018	Marks available	2	Reference code	18M.2.AHL.TZ2.H_9
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Find	Question number	H_9	Adapted from	N/A

Question

The number of taxis arriving at Cardiff Central railway station can be modelled by a Poisson distribution. During busy periods of the day, taxis arrive at a mean rate of 5.3 taxis every 10 minutes. Let T represent a random 10 minute busy period.

Find the probability that exactly 4 taxis arrive during T.

[2]

a.i.

Find the most likely number of taxis that would arrive during T.

[2]

a.ii.

Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.

[3]

a.iii.

During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.

[6]

Markscheme

$X \sim {\text{Po}}\left( {5.3} \right)$

${\text{P}}\left( {X = 4} \right) = {{\text{e}}^{ - 5.3}}\frac{{{{5.3}^4}}}{{4{\text{!}}}}$ (M1)

= 0.164 A1

[2 marks]

a.i.

METHOD 1

listing probabilities (table or graph) M1

mode X = 5 (with probability 0.174) A1

Note: Award M0A0 for 5 (taxis) or mode = 5 with no justification.

METHOD 2

mode is the integer part of mean R1

E(X) = 5.3 ⇒ mode = 5 A1

Note: Do not allow R0A1.

[2 marks]

a.ii.

attempt at conditional probability (M1)

$\frac{{{\text{P}}\left( {X = 7} \right)}}{{{\text{P}}\left( {X \geqslant 6} \right)}}$ or equivalent $\left( { = \frac{{0.1163 \ldots }}{{0.4365 \ldots }}} \right)$ A1

= 0.267 A1

[3 marks]

a.iii.

METHOD 1

the possible arrivals are (2,0), (1,1), (0,2) (A1)

$Y \sim {\text{Po}}\left( {0.65} \right)$ A1

attempt to compute, using sum and product rule, (M1)

0.070106… × 0.52204… + 0.026455… × 0.33932… + 0.0049916… × 0.11028… (A1)(A1)

Note: Award A1 for one correct product and A1 for two other correct products.

= 0.0461 A1

[6 marks]

METHOD 2

recognising a sum of 2 independent Poisson variables eg Z = X + Y R1

$\lambda = 5.3 + \frac{{1.3}}{2}$

P(Z = 2) = 0.0461 (M1)A3

[6 marks]

Examiners report

[N/A]

a.i.

[N/A]

a.ii.

[N/A]

a.iii.

[N/A]

Syllabus sections

Topic 4—Statistics and probability » SL 4.8—Binomial distribution

Show 94 related questions

18M.2.AHL.TZ2.H_8b:
It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures.

Determine the value of n and the value of p.
22M.2.SL.TZ1.9b:
In a random selection of $12$ chocolate muffins, find the probability that exactly $5$ weigh less than $61 g$ .
19M.2.AHL.TZ1.H_9a.ii:
only a sandwich.
19M.2.AHL.TZ1.H_9c.i:
A customer is selected at random. Find the probability that the customer is male and buys a sandwich.
19M.2.AHL.TZ1.H_9c.ii:
A female customer is selected at random. Find the probability that she buys a sandwich.
19M.2.AHL.TZ2.H_2a:
Calculate the probability that, on a randomly selected day, Timmy makes a profit.
18M.2.AHL.TZ2.H_8a:
Find the least possible value of n.
18N.2.SL.TZ0.S_9c:
A randomly selected participant has a reaction time greater than 0.65 seconds. Find the probability that the participant is in Group X.
18N.2.SL.TZ0.S_9a:
A person is selected at random from Group X. Find the probability that their reaction time is greater than 0.65 seconds.
18M.2.SL.TZ2.S_10d.i:
Find the expected number of bags in this crate that contain at most one small apple.
19M.2.AHL.TZ2.H_10b:
Copy and complete the probability distribution table for Y.
SPM.2.SL.TZ0.4b:
The die is tossed five times. Find the probability of obtaining the third “six” on the fifth toss.
17N.2.SL.TZ0.S_8c:
Write down the number of low production hives.
19M.2.SL.TZ1.S_10c.i:
Write down the value of $x$ .
18M.2.SL.TZ2.S_10d.ii:
Find the probability that at least 48 bags in this crate contain at most one small apple.
19M.2.AHL.TZ1.H_9b.ii:
Find the probability that more than 100 cakes will be sold on a typical day.
SPM.2.SL.TZ0.8b.ii:
Hence, find the probability that a seashell selected at random has a length greater than 26 mm.
19N.2.SL.TZ0.S_9a:
Calculate the probability a flight is not late.
19M.2.SL.TZ1.S_10c.ii:
Hence, find the value of $y$ .
17N.2.SL.TZ0.S_8e:
Adam decides to increase the number of bees in each low production hive. Research suggests that there is a probability of 0.75 that a low production hive becomes a regular production hive. Calculate the probability that 30 low production hives become regular production hives.
18M.2.AHL.TZ2.H_9a.ii:
Find the most likely number of taxis that would arrive during T.
17M.2.SL.TZ1.S_4b.ii:
Hence, find the probability that fewer than $k$ students are left handed.
18N.2.SL.TZ0.S_9d:
Ten of the participants with reaction times greater than 0.65 are selected at random. Find the probability that at least two of them are in Group X.
18M.2.SL.TZ2.S_10c:
Find the probability that a bag of apples selected at random contains at most one small apple.
18M.2.SL.TZ1.S_9b.ii:
Hence, find the value of σ.
17M.2.AHL.TZ2.H_10a:
Show that $a = 32$ and $b = \frac{1}{{12}}$ .
18M.2.SL.TZ2.S_10a.ii:
Show that μ = 106.
17N.2.SL.TZ0.S_8d.ii:
Find the number of hives that have a high production.
18M.2.SL.TZ1.S_9b.i:
Find the standardized value for 289 g.
18M.2.SL.TZ2.S_10a.i:
Write down the value of k.
17N.2.SL.TZ0.S_8a:
Write down the value of $a$ and of $b$ .
19M.2.AHL.TZ1.H_9a.i:
both a sandwich and a cake.
17M.2.AHL.TZ2.H_10f:
Find ${\text{P}}(Y \geqslant 3)$ .
19N.2.SL.TZ0.S_9c.ii:
Given that at least $7$ of these flights are on time, find the probability that exactly $10$ flights are on time.
19N.2.SL.TZ0.S_9b:
Find the value of $m$ .
17M.2.AHL.TZ2.H_10c:
Find ${\text{Var}}(X)$ .
18N.2.SL.TZ0.S_9b:
The probability that the reaction time of a person in Group Y is greater than 0.65 seconds is 0.396. Find the value of σ.
19M.2.AHL.TZ2.H_10d:
In any given year of 365 days, the probability that Steffi does not visit Will for at most $n$ days in total is 0.5 (to one decimal place). Find the value of $n$ .
18M.2.AHL.TZ2.H_9b:
During quiet periods of the day, taxis arrive at a mean rate of 1.3 taxis every 10 minutes.

Find the probability that during a period of 15 minutes, of which the first 10 minutes is busy and the next 5 minutes is quiet, that exactly 2 taxis arrive.
18M.2.AHL.TZ1.H_6:
The mean number of squirrels in a certain area is known to be 3.2 squirrels per hectare of woodland. Within this area, there is a 56 hectare woodland nature reserve. It is known that there are currently at least 168 squirrels in this reserve.

Assuming the population of squirrels follow a Poisson distribution, calculate the probability that there are more than 190 squirrels in the reserve.
22M.2.SL.TZ2.9c:
Find the probability that she will be late to work at least one day next week.
22M.2.SL.TZ2.9d:
Given that Suzi will be late to work at least one day next week, find the probability that she will be late less than three times.
22M.2.SL.TZ2.9e:
Find the probability that Suzi will receive a bonus.
20N.2.SL.TZ0.T_6a:
Sketch a diagram to represent this information.
20N.2.SL.TZ0.T_6b:
Show that $m = 15.7$ .
20N.2.SL.TZ0.T_6c.i:
Find the probability that Emlyn plays between $13 minutes$ and $18 minutes$ in a game.
20N.2.SL.TZ0.T_6d:
Find the value of $x$ .
20N.2.SL.TZ0.T_6f:
Find the expected number of successful shots Emlyn will make on Monday, based on the results from Saturday and Sunday.
19N.2.SL.TZ0.S_9d:
SpeedWay increases the number of flights from city ${\text{A}}$ to city ${\text{B}}$ to $20$ flights each week, and improves their efficiency so that more flights are on time. The probability that at least $19$ flights are on time is $0.788$ .

A flight is chosen at random. Calculate the probability that it is on time.
16N.2.SL.TZ0.S_7b:
Let $X$ be the number of red discs selected. Find the smallest value of $m$ for which ${\text{Var}}(X{\text{ }}) < 0.6$ .
18M.2.SL.TZ1.S_9d:
Find the probability that the grocer buys more than half the oranges in a box selected at random.
SPM.2.SL.TZ0.8d:
Find the probability that exactly three of these seashells have a length greater than 26 mm.
18M.2.AHL.TZ2.H_9a.iii:
Given that more than 5 taxis arrive during T, find the probability that exactly 7 taxis arrive during T.
16N.2.AHL.TZ0.H_3b:
Given that ${\text{P}}(X = 2) = 0.241667$ and ${\text{P}}(X = 3) = 0.112777$ , use part (a) to find the value of $\mu$ .
17N.1.AHL.TZ0.H_10b.i:
Determine the mean of X.
SPM.2.SL.TZ0.8a:
Find P(24.15 < X < 25).
16N.2.SL.TZ0.S_7a:
Write down the probability that the first disc selected is red.
19M.2.AHL.TZ2.H_2b:
The shop is open for 24 days every month.

Calculate the probability that, in a randomly selected month, Timmy makes a profit on between 5 and 10 days (inclusive).
SPM.2.SL.TZ0.8c:
Find E(Y).
19M.2.SL.TZ1.S_10a.ii:
Find the probability of rolling two or more red faces.
19M.2.AHL.TZ2.H_10c:
Hence find the expected number of times per day that Steffi is fed at Will’s house.
17M.2.AHL.TZ2.H_10d:
Find the median of $X$ .
17N.1.AHL.TZ0.H_10a:
Show that the probability that Chloe wins the game is $\frac{3}{8}$ .
17N.2.SL.TZ0.S_8d.i:
Find the value of $k$ ;
18M.2.SL.TZ1.S_9c:
To the nearest gram, find the minimum weight of an orange that the grocer will buy.
19N.2.SL.TZ0.S_9c.i:
Calculate the probability that at least $7$ of these flights are on time.
17M.2.AHL.TZ2.H_10b:
Find ${\text{E}}(X)$ .
SPM.2.SL.TZ0.8b.i:
Find $\sigma$ , the standard deviation of X.
19M.2.SL.TZ1.S_10b:
Show that, after a turn, the probability that Ted adds exactly $10 to his winnings is ${\frac{1}{3}}$ .
21M.2.SL.TZ2.8e:
Find the probability that more than $6$ of the flights on this particular day will have a flight time of more than $80$ minutes.
SPM.2.SL.TZ0.8e:
A seashell selected at random has a length less than 26 mm.

Find the probability that its length is between 24.15 mm and 25 mm.
22M.2.AHL.TZ2.8:
Rachel and Sophia are competing in a javelin-throwing competition.

The distances, $R$ metres, thrown by Rachel can be modelled by a normal distribution with mean $56.5$ and standard deviation $3$ .

The distances, $S$ metres, thrown by Sophia can be modelled by a normal distribution with mean $57.5$ and standard deviation $1.8$ .

In the first round of competition, each competitor must have five throws. To qualify for the next round of competition, a competitor must record at least one throw of $60$ metres or greater in the first round.

Find the probability that only one of Rachel or Sophia qualifies for the next round of competition.
20N.2.SL.TZ0.T_6c.ii:
Find the probability that Emlyn plays more than $20 minutes$ in a game.
21N.2.SL.TZ0.4b:
Given that there is at least one defective lamp in the sample, find the probability that there are at most two defective lamps.
19M.2.AHL.TZ1.H_9b.i:
Find the expected number of cakes sold on a typical day.
18M.2.SL.TZ1.S_9a:
Find the probability that an orange weighs between 289 g and 310 g.
SPM.2.SL.TZ0.4a:
The die is tossed five times. Find the probability of obtaining at most three “sixes”.
17N.1.AHL.TZ0.H_10b.ii:
Determine the variance of X.
19M.2.SL.TZ1.S_10a.i:
Find the probability of rolling exactly one red face.
17M.2.AHL.TZ2.H_10e:
Find ${\text{E}}(Y)$ .
17M.2.SL.TZ1.S_4a:
Find $k$ .
18M.2.SL.TZ1.S_9e:
The grocer selects two boxes at random.

Find the probability that the grocer buys more than half the oranges in each box.
16N.2.AHL.TZ0.H_3a:
Show that ${\text{P}}(X = x + 1) = \frac{\mu }{{x + 1}} \times {\text{P}}(X = x),{\text{ }}x \in \mathbb{N}$ .
19M.2.SL.TZ1.S_10d:
Ted will always have another turn if he expects an increase to his winnings.

Find the least value of $w$ for which Ted should end the game instead of having another turn.
17N.2.SL.TZ0.S_8b:
Use this regression line to estimate the monthly honey production from a hive that has 270 bees.
21M.2.SL.TZ2.8d:
Find the expected number of flights that will have a flight time of more than $80$ minutes.
20N.2.SL.TZ0.T_6e:
Find the probability he plays between $13 minutes$ and $18 minutes$ in one game and more than $20 minutes$ in the other game.
20N.2.SL.TZ0.T_6g:
Emlyn claims the results from Saturday and Sunday show that his expected number of successful shots will be more than Johan’s.

Determine if Emlyn’s claim is correct. Justify your reasoning.
17M.2.SL.TZ1.S_4b.i:
Hence, find the probability that exactly $k$ students are left handed;
19M.2.AHL.TZ2.H_10e:
Show that the expected number of occasions per year on which Steffi visits Will’s house and is not fed is at least 30.
EXN.2.SL.TZ0.8d:
Find the probability that at least five of them will take more than $30$ minutes to complete the jigsaw puzzle.
18M.2.SL.TZ2.S_10b:
Find P(M < 95) .
19M.2.AHL.TZ2.H_10a:
Find the probability that on a randomly selected day, Steffi does not visit Will’s house.
21N.2.SL.TZ0.4a:
Find the probability that there is at least one defective lamp in the sample.

Hide related questions

Topic 4—Statistics and probability

Question

Markscheme

Examiners report

Syllabus sections

View options